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Torus Actions and Their Applications in Topology and Combinatorics

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Victor M. Buchstaber; Taras E. Panov

The book presents the study of torus actions on topological spaces is presented as a bridge
connecting combinatorial and convex geometry with commutative and homological
algebra, algebraic geometry, and topology. This established link helps in
understanding the geometry and topology of a space with torus action by
studying the combinatorics of the space of orbits. Conversely, subtle
properties of a combinatorial object can be realized by interpreting it as the
orbit structure for a proper manifold or as a complex acted on by a torus. The
latter can be a symplectic manifold with Hamiltonian torus action, a toric
variety or manifold, a subspace arrangement complement, etc., while the
combinatorial objects include simplicial and cubical complexes, polytopes, and
arrangements. This approach also provides a natural topological interpretation
in terms of torus actions of many constructions from commutative and
homological algebra used in combinatorics.

The exposition centers around the theory of moment-angle complexes,
providing an effective way to study invariants of triangulations by methods of
equivariant topology. The book includes many new and well-known open problems
and would be suitable as a textbook. It will be useful for specialists both in
topology and in combinatorics and will help to establish even tighter
connections between the subjects involved.

Readership

Graduate students and research mathematicians interested in
topology or combinatorics; topologists interested in combinatorial applications
and vice versa.

Reviews & Endorsements

The book is quite well-written and includes many new and well-known open
problems

-- Mathematical Reviews

The text contains a wealth of material and … the book may be a
welcome collection for researchers in the field and a useful overview
of the literature for novices.

The book presents the study of torus actions on topological spaces is presented as a bridge
connecting combinatorial and convex geometry with commutative and homological
algebra, algebraic geometry, and topology. This established link helps in
understanding the geometry and topology of a space with torus action by
studying the combinatorics of the space of orbits. Conversely, subtle
properties of a combinatorial object can be realized by interpreting it as the
orbit structure for a proper manifold or as a complex acted on by a torus. The
latter can be a symplectic manifold with Hamiltonian torus action, a toric
variety or manifold, a subspace arrangement complement, etc., while the
combinatorial objects include simplicial and cubical complexes, polytopes, and
arrangements. This approach also provides a natural topological interpretation
in terms of torus actions of many constructions from commutative and
homological algebra used in combinatorics.

The exposition centers around the theory of moment-angle complexes,
providing an effective way to study invariants of triangulations by methods of
equivariant topology. The book includes many new and well-known open problems
and would be suitable as a textbook. It will be useful for specialists both in
topology and in combinatorics and will help to establish even tighter
connections between the subjects involved.